Entropies of non positively curved metric spaces

TL;DR

This paper explores various entropy concepts in non-positively curved metric spaces, establishing their equivalences and providing estimates for boundary subsets in hyperbolic spaces with convex geodesic bicombings.

Contribution

It introduces new equivalences of entropy notions in convex geodesic bicombing spaces and extends estimates to hyperbolic space boundaries.

Findings

Equivalence of topological entropy and boundary Minkowski dimension.

Estimates for boundary subsets in Gromov-hyperbolic spaces.

Applicability to spaces with convex geodesic bicombings.

Abstract

We show the equivalences of several notions of entropy, like a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform packing condition. Similar estimates will be given in case of closed subsets of the boundary of Gromov-hyperbolic metric spaces with convex geodesic bicombings.